Fink's conjecture on ample numbers avoiding the first primes

Let $a(n)$ denote the recursive divisor function, and call a positive integer $n$ ample when $a(n)>n$. Let $p_1,p_2,\ldots$ be the primes in increasing order. Fink's conjecture. For every positive integer $k$, there exists an ample number not divisible by any of $p_1,\ldots,p_k$. This parallels the corresponding known property of abundant numbers, while odd ample numbers and examples avoiding the first prime have been found; the conjecture remains open.